●Wonderful varieties of type B and C Authors: P. Bravi, G. Pezzini (Submitted on 21 Sep 2009) Abstract: We show that the proof of Luna's conjecture about the classification of general wonderful G-varieties can be reduced to the analysis of finitely many families of primitive cases. We work out all primitive cases arising with any classical group G. 24 pages http://arxiv.org/abs/0909.3771
●Deformations of the Lie algebra o(5) in characteristics 3 and 2 Authors: Sofiane Bouarroudj, Alexei Lebedev, Friedrich Wagemann (Submitted on 19 Sep 2009) Abstract: The finite dimensional simple modular Lie algebras with Cartan matrix cannot be deformed if the characteristic p of the ground field is equal to 0 or greater than 3. If p=3, the orthogonal Lie algebra o(5)is one of the two simple modular Lie algebras with Cartan matrix that have deformations (the Brown algebras br(2; a) are among these 10-dimensional deforms and hence are not counted separately); the 29-dimensional Brown algebra br(3) is the only other simple Lie algebra with Cartan matrix that has deformations. Kostrikin and Kuznetsov described the orbits (isomorphism classes) under the action of the group O(5) of automorphisms of o(5) on the space H^2(o(5);o(5)) and produced representatives of the isomorphism classes. Here we explicitly describe global deforms of o(5) and of the simple analog of this orthogonal Lie algebra in characteristic 2. 12 pages, no figures http://arxiv.org/abs/0909.3572
●Primitive spherical systems Authors: P. Bravi (Submitted on 21 Sep 2009) Abstract: A spherical system is a combinatorial object, arising in the theory of wonderful varieties, defined in terms of a root system. All spherical systems can be obtained by means of some general combinatorial procedures (parabolic induction, fiber product and projective fibration) from the so-called primitive spherical systems. Here we report the list of all primitive spherical systems. Comments: 34 pages
●Euler characteristics and compact p-adic Lie groups Authors: Simon Wadsley (Submitted on 21 Sep 2009) Abstract: We discuss Euler characteristics for finitely generated modules over Iwasawa algebras. We show that the Euler characteristic of a module is well-defined whenever the 0th homology group is finite if and only if the relevant compact p-adic Lie group is finite-by-nilpotent and that in this case all pseudo-null modules have trivial Euler characteristic. We also prove some other results relating to the triviality of Euler characteristics for pseudo-null modules. Comments: 9 pages
●Quiver grassmannians, quiver varieties and the preprojective algebra Authors: Alistair Savage, Peter Tingley (Submitted on 21 Sep 2009)
Abstract: Quivers play an important role in the representation theory of algebras, with a key ingredient being the path algebra and the preprojective algebra. Quiver grassmannians are varieties of submodules of a fixed module of the path or preprojective algebra. In the current paper, we study these objects in detail. We show that the quiver grassmannians corresponding to submodules of certain injective modules are homeomorphic to the lagrangian quiver varieties of Nakajima which have been well studied in the context of geometric representation theory. We then refine this result by finding quiver grassmannians which are homeomorphic to the Demazure quiver varieties introduced by the first author, and others which are homeomorphic to the graded/cyclic quiver varieties defined by Nakajima. The Demazure quiver grassmannians allow us to describe injective objects in the category of locally nilpotent modules of the preprojective algebra. We conclude by relating our construction to a similar one of Lusztig using projectives in place of injectives. Comments: 27 pages
●Cohomology of finite dimensional pointed Hopf algebras Authors: M. Mastnak, J. Pevtsova, P. Schauenburg, S. Witherspoon (Submitted on 4 Feb 2009 (v1), last revised 19 Sep 2009 (this version, v2))
Abstract: We prove finite generation of the cohomology ring of any finite dimensional pointed Hopf algebra, 私たちは、任意の有限次元の pointed ホップ代数から、有限次数のコホモロジー環を生成することに成功した。 having abelian group of grouplike elements, under some mild restrictions on the group order. 擬群の要素としてのアーベル群を持つ、群の条件を緩和して。 The proof uses the recent classification by Andruskiewitsch and Schneider of such Hopf algebras. この証明は、Andruskiewitsch and Schneiderによる最近のホップ代数の分類を用いている。 Examples include all of Lusztig's small quantum groups, whose cohomology was first computed explicitly by Ginzburg and Kumar, 例は、Lusztigの量子群を含んでいる、そのコホモロジーはGinzburg and Kumarによって初めて計算された、 as well as many new pointed Hopf algebras. その他の指摘されたホップ代数同様に。 We also show that in general the cohomology ring of a Hopf algebra in a braided category is braided commutative. 組み紐のカテゴリーにあるホップ代数のコホモロジー環は、組み紐論的に可換であることが示される。 As a consequence we obtain some further information about the structure of the cohomology ring of a finite dimensional pointed Hopf algebra and its related Nichols algebra. 結論として、有限次元のpointed ホップ代数のコホモロジー環について進んだ情報を得ることができた。そして、Nichols環にも関係しているということを。 Comments: 36 pages, references added
●Undergraduate Lecture Notes in De Rham-Hodge Theory Authors: Vladimir G. Ivancevic, Tijana T. Ivancevic (Submitted on 31 Jul 2008 (v1), last revised 21 Oct 2008 (this version, v3)) 19 pages, 4 figures, Latex http://arxiv.org/abs/0807.4991
●Dirac operator on the restricted Grassmannian manifold Authors: Vesa Tahtinen 制約されたグラスマン多様体上におけるディラック作用素について ヴェサ・ターティネン (Submitted on 7 Sep 2009 (v1), last revised 20 Sep 2009 (this version, v2))
Abstract: In his book Mickelsson notices that the infinite-dimensional Grassmannian manifold of Segal and Wilson マイケルソンの著書の中で、セーガルとウィルソンの有限次元のグラスマン多様体は admits a Spin^c structure and after this he naturally considers the problem of defining a Dirac operator on it. スピンC構造を許すので、その上にディラック作用素が考えられないかどうかと提起している。 Mickelsson gives a possible candidate for such an operator but unfortunately it proves out to be badly diverging and しかし、残念ながらそれは発散することが示され he leaves it as an open problem to introduce proper modifications to his original construction in order to obtain a well-defined (unbounded) operator with expected properties. 彼はそれを公開の未解決問題に指定した。 Using fermionic Fock space representations of the restricted unitary group associated to a polarized Hilbert space, 制約付きのユニタリ群におけるフェルミオン-フォック空間の表現は、極化されたヒルベルト空間と関係があり introduced in the well-known book written by Pressley and Segal on loop groups, Pressley and Segalによって書かれた、ループ群に関する有名な本に紹介された。 we construct a well-defined candidate for a Dirac operator on the restricted Grassmannian manifold acting on a relevant space of spinors. ウェルディファインドなディラク作用素を提起する。 As our main result we show that our operator is an unbounded symmetric operator with finite-dimensional kernel. 結果:対称作用素、有限次元の核付き。 81 pages, corrected references
●Multiplet containing components with different masses Authors: D.V. Soroka, V.A. Soroka (Submitted on 20 Sep 2009) Abstract: A principle possibility for the existence of a multiplet including the components with the different masses is indicated. This paper is dedicated to the memory of Anna Yakovlevna Gelyukh (Kalaida). http://arxiv.org/abs/0909.3624 Comments: 6 pages, Latex. A contribution to the Proceedings of the International Workshop "Supersymmetries and Quantum Symmetries" (SQS'09) July 29 - August 3, 2009, Dubna, Russia
De GL(2,F) a Gal_{Q_p} Authors: Marie-France Vigneras (Submitted on 22 Sep 2009) Abstract: We construct a functor from the category of admissible finitely presented o-representations of GL(2,F) to the category of finite length o-representations of Gal_{Q_p}, for any finite extension F of Q_p and the ring of integers o of a finite extension L/Q_p. http://arxiv.org/abs/0909.4003
●Soliton equations, vertex operators, and simple singularities Authors: E. Frenkel, A. Givental, T. Milanov (Submitted on 22 Sep 2009)http://arxiv.org/abs/0909.4032 ソリトン方程式、頂点作用素、単純特異点
Abstract: We prove the equivalence of two hierarchies of soliton equations associated to a simply-laced finite Dynkin diagram. ソリトン方程式の階列と、ディンキン図形の階列との同等性を証明した。 The first was defined by Kac and Wakimoto using the principal realization of the basic representations of the corresponding affine Kac-Moody algebra. 最初は、Kacと脇本実によって 対応するアフィン カッツ・ムーディ代数による基底表現によって定義された。 The second was defined in arXiv:math/0307176 using the Frobenius structure on the local ring of the corresponding simple singularity. 第2に、対応する単純特異点の局所環上のフロベニウス構造として定義された。 We also obtain a deformation of the principal realization of the basic representation over the space of miniversal deformations of the corresponding singularity. As a by-product, we compute the operator product expansions of pairs of vertex operators defined in terms of Picard-Lefschetz periods for more general singularities. Thus, we establish a surprising link between twisted vertex operators and deformation theory of singularities. 特異点の変形理論と、ねじれ頂点作用素との間に存在する関係を確立することができた。
●Constructing representations of Hecke algebras for complex reflection groups Authors: Gunter Malle, Jean Michel (IMJ) (Submitted on 22 Sep 2009) http://arxiv.org/abs/0909.4040 複素射影群のためのヘッケ代数の構成表現 Abstract: We investigate the representations and the structure of Hecke algebras associated to certain finite complex reflection groups. ある有限次複素射影群におけるヘッケ代数の表現を探索した。 We first describe computational methods for the construction of irreducible representations of these algebras, including a generalization of the concept of $W$-graph to the situation of complex reflection groups.
We then use these techniques to find models for all irreducible representations in the case of complex reflection groups of dimension at most three. Using these models we are able to verify some important conjectures on the structure of Hecke algebras.
●The structure of parafermion vertex operator algebras: general case Authors: Chongying Dong, Qing Wang (Submitted on 21 Sep 2009) http://arxiv.org/abs/0909.3872 パラフェルミオンにおける頂点作用素代数;一般論 Abstract: The structure of the parafermion vertex operator algebra associated to an integrable highest weight module for any affine Kac-Moody algebra is studied. パラフェルミオンにおける頂点作用素代数は、アフィン カッツ・ムーディ代数の極大モジュールと関連している。 In particular, a set of generators for this algebra has been determined. 特に、生成素が決定される。 Comments: 15 pages
●Quivers with potentials associated to triangulated surfaces, Part II: Arc representations Authors: Daniel Labardini-Fragoso (Submitted on 23 Sep 2009) http://arxiv.org/abs/0909.4100 Abstract: This paper is a representation-theoretic extension of Part I. It has been inspired by three recent developments: surface cluster algebras studied by Fomin-Shapiro-Thurston, the mutation theory of quivers with potentials initiated by Derksen-Weyman-Zelevinsky, and string modules associated to arcs on unpunctured surfaces by Assem-Brustle-Charbonneau-Plamondon. Modifying the latter construction, to each arc and each ideal triangulation of a bordered marked surface we associate in an explicit way a representation of the quiver with potential constructed in Part I, so that whenever two ideal triangulations are related by a flip, the associated representations are related by the corresponding mutation. Comments: 51 pages; 37 figures
●The linkage principle for restricted critical level representations of affine Kac-Moody algebras Authors: Tomoyuki Arakawa, Peter Fiebig (Submitted on 23 Sep 2009) Abstract: We study the restricted category O for an affine Kac--Moody algebra at the critical level. In particular, we prove the first part of the Feigin-Frenkel conjecture: the linkage principle for restricted Verma modules. Moreover, we prove a version of the BGGH-reciprocity principle and we determine the block decomposition of the restricted category O. For the proofs we need a deformed version of the classical structures, so we mostly work in a relative setting. Comments: 40 pages
●The Quiver of Projectives in Hereditary Categories with Serre Duality Authors: Carl Fredrik Berg, Adam-Christiaan van Roosmalen (Submitted on 9 Jan 2008 (v1), last revised 23 Sep 2009 (this version, v2)) http://arxiv.org/abs/0801.1461
Abstract: Let k be an algebraically closed field and A a k-linear hereditary category satisfying Serre duality with no infinite radicals between the preprojective objects. If A is generated by the preprojective objects, then we show that A is derived equivalent to rep_k Q for a so called strongly locally finite quiver Q. To this end, we introduce light cone distances and round trip distances on quivers which will be used to investigate sections in stable translation quivers of the form \mathbb{Z} Q.
Comments:16 pages, as accepted by Journal of Pure and Applied Algebra
http://simonsingh.net/Fields_Medallist.html 「Borcherds admits that being ignored was partly his own fault. He finds it difficult to communicate, and tends to avoid discussions with others. For example, he prefers to read a published paper, rather than talk to the author, and he no longer teaches or gives tutorials. 」 アスペルガーの人は議論をしたがらないのか?読むだけで、著者には何も言わず、それで終わりか。
「The ‘moonshine’ was added, because the term has long been used to describe absurd scientific ideas. Ernest Rutherford once said that it was moonshine to suggest that we could ever obtain energy from atoms. His wife sometimes claims that he has Asperger’s Syndrome, a very mild form of autism which is characterised by introversion and a lack of emotion.」 嫁さんはいるんだな。
「The ‘moonshine’ was added, because the term has long been used to describe absurd scientific ideas. Ernest Rutherford once said that it was moonshine to suggest that we could ever obtain energy from atoms.」 へぇ〜ラザフォードが最初に“moonshine”という言葉を出したんだね。 現代日本語で考えたら「電波ゆんゆん」、「お花畑」とかなんかそんな感じだよな。 モンスター=ムーンシャインは今や天下の大予想になってしまったのだが。
「He claims that lack of understanding from others does not bother him, and that what really matters is the satisfaction of solving a great problem. 」 他者にどう見られるかっていうのはどっちでもいいこと。本当に大事なのは問題が解けたときの喜びを得られるかどうか、ってか。 すごすぎるわこの人。これが本物の数学者の姿か。。。
「Even the award of a Fields Medal is not important compared to completing an immense calculation, and his reaction to the news was lukewarm at best. “I didn’t really feel anything,” he says. “Before the award I used to think it was terribly important, but now I realise that it’s meaningless. However, I was over the moon when I proved the moonshine conjecture. If I get a good result I spend several days feeling really happy about it. I sometimes wonder if this is the feeling you get when you take certain drugs. I don’t actually know, as I have not tested this theory of mine.”」 ムーンシャイン予想が解けたら、それこそ昇天しそうなくらい喜べる、ってか。 196,883 dimensions, and 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 symmetries よく計算するよな、って思う。もう超人クラスだね。解けたらものすごい数学的成果であることには間違いない。 今も多分やっておられるのだろう。いや、ムーンシャインが解けるまでは死ぬまで取り組まれるであろう。。。
[Ko1] Kottwitz, R. Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992),no. 2, 373?444. [Ko2] Kottwitz, R. On the ,-adic representations associated to some simple Shimura varieties, Invent.Math. 108 (1992), no. 3, 653?665.
整数論に関する参考文献 ●局所類体論 1. 斎藤秀司, 『整数論』, 共立講座21世紀の数学, 共立出版 2. K. Iwasawa, Local class field theory, Oxford Science Publications. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (日本語版を元に,Lubin-Tate理論を中心に使う形に大幅に書き直してある) 3. J.-P. Serre, Corps locaux, Hermann, Paris (英訳 : Local fields, Graduate Texts in Mathematics, 67. Springer-Verlag, New York-Berlin) ●大域類体論 1. 高木貞治, 『代数的整数論』, 岩波書店 2. 加藤和也,斎藤毅,黒川信重, 『数論〈1〉Fermatの夢と類体論』,岩波書店 3. S. Lang, Algebraic number theory, Second edition. Graduate Texts in Mathematics, 110. Springer-Verlag, New York 4. E. Artin, J. Tate, Class field theory, W. A. Benjamin, Inc., New York-Amsterdam 5. J. W. S. Cassels, A. Froehlich, Algebaic Number Theory Academic Press, London and New York 6. A. Weil, Basic number theory, Classics in Mathematics. Springer-Verlag, Berlin ●その他(講義ノートなど) 1. J. S. Milne, Course Notes (Algebraic Number Theory, Class Field Theory 等) (Milneのホームページ) [特に Class Field Theory の講義録はお勧めです] 2. T. Yoshida, Local class field theory via Lubin-Tate theory, preprint, 2006. (arXiv) 3. 吉田輝義, 『GL(n) の大域・局所 Langlands 対応』, 第50回代数学シンポジウム報告集, 2005年. (報告集ページ)
●三角形大全 The Triangle Book (Hardcover) by John H. Conway (Author), Steve Sigur (Author) # Hardcover: 400 pages # Publisher: AK Peters, Ltd.; 1 edition (June 15, 2005) 2009年12月 未刊 ご予約承ります。 アマチュア数学者、高校生たちがプログラミングしまくって出した成果の集大成。 有限群の発見者で有名なConwayは何を書いているのかな。どんな成果があるのかな?面白そう。
●Symplectic Fibrations and Multiplicity Diagrams, Ed. 1 Author : Guillemin, Victor/ Lerman, Eugene/ Sternberg, Shlomo Publisher : CAMBRIDGE UNIVERSITY PRESS Publication Year : 2009
●Cohomological Methods in Transformation Groups, Ed. 1 Author : Allday, C./ Puppe, V.Publisher : CAMBRIDGE UNIVERSITY PRESS Publication Year : 2009
●Low Dimensional Topology Author : Mrowka, Tomasz S. (EDT)/ Ozsvath, Peter S. (EDT) Publisher : AMS Publication Year : 2009
Generalized Moonshine I: Genus zero functions Authors: Scott Carnahan (Submitted on 18 Dec 2008 (v1), last revised 28 Dec 2008 (this version, v2)) http://arxiv.org/abs/0812.3440
Generalized moonshine II: Borcherds products Authors: Scott Carnahan (Submitted on 28 Aug 2009 (v1), last revised 1 Oct 2009 (this version, v2)) http://arxiv.org/abs/0908.4223
Affine Deligne-Lusztig varieties in affine flag varieties Authors: Ulrich Goertz, Thomas J. Haines, Robert E. Kottwitz, Daniel C. Reuman (Submitted on 1 May 2008 (v1), last revised 1 Oct 2009 (this version, v3)) http://arxiv.org/abs/0805.0045
Abstract: This paper studies affine Deligne-Lusztig varieties in the affine flag manifold of a split group. Among other things, it proves emptiness for certain of these varieties, relates some of them to those for Levi subgroups, extends previous conjectures concerning their dimensions, and generalizes the superset method.
Comments: 44 pages, 4 figures. Minor changes to font, references, and acknowledgments. Improved introduction, other improvements in exposition, and two new figures added, for a total of 4